Let $X\sim Poi(10) $ and $Y \sim exp(\frac{1}{10})$ independent. Why $P(X+Y\leq\frac{3}{2}) = P(X=0, Y \leq \frac{3}{2}) + P(X=1,Y\leq \frac {1}{2})$?

Question

Let $X\sim Poi(10) $ and $Y \sim exp(\frac{1}{10})$ independent random raviables

I would like to compute: $P(X+Y\leq\frac{3}{2})$

So what I did, which is probably wrong, is the following:

Let $X=k$ then $P(X+Y\leq\frac{3}{2})=P(X=k, Y\leq\frac{3}{2}-k)$

But I don't know how to take it from there.

The solution on the other hand computes it as follows:

$P(X+Y\leq\frac{3}{2}) = P(X=0, Y \leq \frac{3}{2}) + P(X=1,Y\leq \frac {1}{2})$

But why does $X\in${0,1} ? it doesn't make much sense to me.

Thanks!!

Answer 1

$X$ and $Y$ take only non-negative integer values. $X+Y \leq \frac 3 2$ implies $X\leq \frac 3 2$ and the only integers less than or equal to $\frac 3 2$ are $0$ and $1$.

Answer 2

That's because $X \in \mathbb{N} \cup \{0\}$ and $Y > 0$, so $X + Y$ can only be below $3/2$ when $X = 0$ or $X = 1$.